# A field with $2=0$ [duplicate]

Is there any field other than $\mathbb{F}_2$ which has $1+1=0$ ? My initial feeling was no but I don't know how to prove it. Can someone help ?

• How about $\mathbb{F}_4$? In fact any field of characteristic $2$ would do it. – Stefan4024 Sep 8 '18 at 7:40

This property is called "characteristic $2$", and many fields have it. $\Bbb F_2$ is in some sense the most "basic" one, but we also have examples like $$\Bbb F_4=\Bbb F_2[t]/(t^2+t+1)$$ or $\Bbb F_2(x)$, the field of rational functions in one variable with coefficients from $\Bbb F_2$.
• "in some sense the most basic one" -- why not be precise and say: Every field of characteristic 2 contains a unique copy of $\Bbb F_2$, and conversely, every field which contains $\Bbb F_2$ is of characteristic 2. – Torsten Schoeneberg Sep 9 '18 at 23:58